The MRI in a Collisionless Plasma Eliot Quataert (UC Berkeley) Collaborators: Prateek Sharma, Greg Hammett, Jim Stone
Modes of Accretion thin disk: energy radiated away (relevant to star & planet formation, galaxies, & luminous compact objects) prad >> pgas ionization state & Hall effects last stable orbit & radiative efficiency? thick disk (~ torus): energy stored as heat (relevant to BHs & NSs at very low/high accretion rates) jet production Ti/Te? efficiency? super-eddington?
Radiatively Inefficient Accretion Flows At low densities (accretion rates), cooling is inefficient grav. energy turbulence (MRI) thermal energy: not radiated kt ~ GMmp/R (virial): Tp ~ 10 11-12 K > Te ~ 10 10-11 K near BH collisionless plasma: e-p equil. time > inflow time for Hawley & Balbus 2002 Disk Structure Accretion Rate (Stone & Pringle 2001; Hawley & Balbus 2002; Igumenshchev et al. 2003)
Relevant Physics in the Kinetic Limit Collisionless Damping operates at all k, not just high k Alfven waves also damped (linearly coupled to slow mode via rotation) Free Streaming along B Anisotropic Heat & Momentum Transport Anisotropic Pressure Tensor MFP set by wave-particle interactions, not collisions Collisionless Damping of Fast & Slow MHD Modes at β = 1 Note: must use anisotropic viscosity (not scalar viscosity)
The Linear MRI in Kinetic Theory fastest growth at low k where tension is weak linear instability unchanged for Bz, kz only in general, angular momentum transport via anisotropic pressure (viscosity!) in addition to magnetic stresses anisotropic viscosity is destabilizing, unlike isotropic viscosity Quataert, Dorland, Hammett 2002; also Sharma et al. 2003; Balbus 2004
Nonlinear Evolution Simulated using Kinetic-MHD Large-scale dynamics of collisionless plasmas: expand Vlasov eqn using slow timescale and large lengthscale assumptions of MHD (Kulsrud 1983) more general than the Braginskii eqns Particles efficiently transport heat and momentum along B-field lines
Evolution of the Pressure Tensor adiabatic invariance of μ ~ v 2 /B ~ T /B closure model in sims k ~ L -1 equiv to κ ~ vthl
Shearing Box Sims of the Kinetic MRI magnetic energy stress pressure anisotropy Saturates at Small Amplitudes! w/ δb/b ~ 1/β Box filled w/ stable anisotropic Alfven waves
Pressure Anisotropy µ T /B = constant T > T as B a background pressure anisotropy (p > p ) can stabilize the MRI But... T T unstable to small-scale (Larmor radius) instabilities that act to isotropize the pressure tensor (velocity space instabilities) e.g., mirror, firehose, ion cyclotron, electron whistler Fluctuations have ω ~ cyclotron: violate μ invariance & pitch angle scatter Use subgrid model to account for this physics in Shearing Box Sims p t p t =... ν(p, p,β)[ p p ] =... ν(p, p,β)[ p p ]
Velocity-Space Instabilities in the Solar Wind Electron Anisotropy Te, /Te, whistler proton anisotropy limited by firehose Kasper et al. 2002 Stuart Bale firehose βe,
Shearing Box Sims of the Kinetic MRI (w/ pitch angle scattering & mean Bz) magnetic energy volume averaged pressure anisotropy Δp/p ~ few % Anisotropic Stress ~ Maxwell Stress
The Approach to MHD Nonlinear Evolution for Difft. Scattering Rates ν magnetic stress anisotropic stress collision frequency MHD is reached when ν/ω > 10 ~ β 1/2
Energetics in the (Kinetic) Shearing Box Shear Anisotropic Pressure Turbulence (MRI) Grid-scale Loss of Magnetic & Kinetic Energy (~ 1/2 of GPE) Collisionless Damping of Fluctuations (small) Direct Viscous Heating on Resolved Scales (~ 1/2 of Grav. Pot. Energy)
Heating ~ Shear*Stress Collisional Plasma Coulomb Collisions set Δp q + ~ m 1/2 T 5/2 Primarily Ion Heating Collisionless Plasma Microinstabilities Regulate Δp Significant Electron Heating
Ti/Te in Two-Species Kinetic MRI Sims Ti/Te ~ 10 at late times
Astrophysical Implications Lines w/ smaller T at a given R correspond to larger accretion rate best guess Line of fixed L = 10 36 erg/s for Sgr A* Use q + ~ T 1/2 in 1D models to determine Te(r) & radiation Sgr A*: Predicted consistent w/ Faraday Rotation & measured Te from VLBI
Summary MRI growth is enhanced in a collisionless plasma anisotropic viscosity is destabilizing so Pm effects fundamentally difft. Nonlinear amplification & saturation requires isotropization of the plasma pressure by small-scale kinetic instabilities Anisotropic Stress ~ Maxwell Stress ~ 1/2 of the grav. pot. energy thermalized on large scales significant electron heating